Polynomial Inequalities in One Variable

Let P(x) be any polynomial. Then P(x) < 0 and P(x) > 0 are called polynomial inequalities.

To solve polynomial inequalities:

Use a sign graph of P(x)

Analyze a graph of P(x):

P(x) > 0 when the graph is above the x-axis

P(x) < 0 when the graph is below the x-axis

3 2Example 1. Solve 2 3 0 by using a sign graph.x x x

Find the zeros of the polynomial 3 22 3P x x x x

2 2 3P x x x x

1 3P x x x x 1, 0, 3x x x

Plot the zeros on a number line

Test for the signs of P(x) in each interval

2P

1

2P

1P

4P

The solution of P(x) < 0 is x < -1 or 0 < x < 3

ActivityActivity

2a. What is the sign of 1 for 1? for 1?x x x

2b. Explain why 1 does not change sign at 1.x x

2c. What is the sign of the product 2 3 1 for 3 1?x x x

2What is the sign of the product 2 3 1 for 1?x x x Explain why this product should not change sign at x = 1.

Not all polynomials change sign at a zero.

2

A polynomial will not change sign at a zero

if corresponds to the squared factor .

P x c

c x c

22Example 2. Solve 1 4 0.x x Find the zeros: 1, 1, 4x x x double root

21 1 4P x x x x

22P

20P

22P

25P

Example 3. Solve 22 5

04

x x

x

Find the zeros of all linear factors in the numerator and denominator:

2, 4, 5, 4x x x x

4 52

22 5

04

x xf x

x

2

0f

2

3f

2

4.5f

2

6f

The solution of f(x) < 0 is –2 < x < 4 or x = 5

Example 4. Solve 3 22 8 3 0x x x

by using a graphing calculator.